On Gradient Descent Algorithm for Generalized Phase Retrieval Problem

Understand

In this paper, we study the generalized phase retrieval problem: to recover a signal $\bm{x}\in\mathbb{C}^n$ from the measurements $y_r=\lvert \langle\bm{a}_r,\bm{x}\rangle\rvert^2$, $r=1,2,\ldots,m$.

• The problem can be reformulated as a least-squares minimization problem.
• Although the cost function is nonconvex, the global convergence of gradient descent algorithm from a random initialization is studied, when $m$ is large enough.
• We improve the known result of the local convergence from a spectral initialization.